Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6347597 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #62729 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
2942 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,6)(4,7)(5,8)(9,15)(10,13)(11,14)(12,16)(17,18,23,27)(19,28,31,38)(20,25,33,41)(21,26,22,35)(24,36,39,30)(29,42,44,47)(32,40,34,45)(37,46,48,43);(1,3)(2,6)(4,9)(5,10)(7,12)(8,13)(11,16)(14,15)(17,19)(18,24)(20,29)(21,30)(22,36)(23,31)(25,37)(26,38)(27,39)(28,35)(32,43)(33,44)(34,46)(40,47)(41,48)(42,45);(1,4)(2,7)(3,9)(5,11)(6,12)(8,14)(10,16)(13,15)(17,20)(18,25)(19,29)(21,32)(22,34)(23,33)(24,37)(26,40)(27,41)(28,42)(30,43)(31,44)(35,45)(36,46)(38,47)(39,48);(2,8)(6,13)(7,14)(12,15)(17,21,23,22)(18,26,27,35)(19,30,31,36)(20,32,33,34)(24,38,39,28)(25,40,41,45)(29,43,44,46)(37,47,48,42);(20,33)(25,41)(29,44)(32,34)(37,48)(40,45)(42,47)(43,46)Polycyclic presentation:
2^9|a^2=h,d^2=h,f^2=h,b^a=bf,c^a=cg,d^a=di,f^a=fh,f^b=fh,g^b=gi,e^c=eh,f^c=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6254798);